CBSE Class 11 Mathematics HOTs Conic Sections

Question. If (a, b) is a point on the circle whose centre is on the x-axis and which touches the line x + y = 0 at (2, –2), then the greatest value of a is
(a) 4 – √2
(b) 6
(c) 4 + 2√2
(d) 4 + √2
Answer : C

Question. A ray of light incident at the point (– 2, – 1) gets reflected from the tangent at (0, –1) to the circle x² + y² = 1. The reflected ray touches the circle. The equation the line along which the incident ray moved, is
(a) 4x - 3y +11 = 0
(b) 4x + 3y +11 = 0
(c) 3x + 4y +11 = 0
(d) 4x + 3y + 7 = 0
Answer : B

Question. The set of all real values of l for which exactly two common tangents can be drawn to the circles x² + y² – 4x – 4y + 6 = 0 and x² + y² – 10x – 10y + λ = 0 is the interval: 
(a) (12, 32)
(b) (18, 42)
(c) (12, 24)
(d) (18, 48)
Answer : B

Question. The set of values of ‘c’ so that the equations y = | x | + c and x² + y² – 8| x | – 9 = 0 have no solution is
(a) (– ∞ , – 3) ∪ (3, ∞)
(b) (– 3, 3)
(c) (– ∞, – √2 ) ∪ (5 √2 , ∞)
(d) (5 √2 –4, ∞)
Answer : D

Question. Tangent to the curve y = x² + 6 at a point (1, 7) touches the circle x² + y² + 16x + 12y + c = 0 at a point Q. Then the coordinates of Q are
(a) (–6, –11)
(b) (–9, –13)
(c) (–10, –15)
(d) (–6, –7)
Answer : D

Question. If the line y = mx + 1 meets the circle x² + y² + 3x = 0 in two points equidistant from and on opposite sides of x-axis, then
(a) 3m + 2 = 0
(b) 3m – 2 = 0
(c) 2m + 3 = 0
(d) 2m – 3 = 0
Answer : B

Question. The line 4x + 3y - 4 = 0 divides the circumference of the circle centered at (5, 3), in the ratio 1 : 2. Then the equation of the circle is
(a) x² + y² -10x - 6y - 66 = 0
(b) x² + y² -10x - 6y +100 = 0
(c) x² + y² -10x - 6y + 66 = 0
(d) x² + y² -10x - 6y –100 = 0
Answer : A

Question. Let A(– 4, 0) and B(4, 0). Then the number of points C = (x, y) on the circle x² + y² = 16 lying in first quadrant such that the area of the triangle whose vertices are A, B and C is a integer is
(a) 14
(b) 15
(c) 16
(d) None of these
Answer : B

Question. If a circle passes through the point (a, b) and cuts the circle x² + y² = 4 orthogonally, then the locus of its centre is
(a) 2ax - 2by - (a² + b² + 4) = 0
(b) 2ax + 2by - (a² + b² + 4) = 0
(c) 2ax - 2by + (a² + b² + 4) = 0
(d) 2ax + 2by + (a² + b² + 4) = 0
Answer : B

Question. A circle is drawn with centre at the focus S of the parabola y² = 4x so that a common chord of the parabola and the circle is equidistant from the focus and the vertex. Then the equation of the circle is
(a) (x – 1)² + y² = 9/4
(b) (x – 1)² =  9/16 – y²
(c) (x – 1)² + x² = 9/4
(d) (y – 1)² + x² = 9/16
Answer : A

Question. Locus of all such points so that sum of its distances from (2, – 3) and (2, 5) is always 10, is
(a) (x - 2)² / 25 + (y - 1)² / 9 = 1
(b) (x - 2)² / 25 + (y - 1)² / 16 = 1
(c) (x - 2)² / 16 + (y - 1)² / 25 = 1
(d) (x - 2)² / 9 + (y - 1)² / 25 = 1
Answer : D

Question. Let L₁ be the length of the common chord of the curves x² + y² = 9 and y² = 8x, and L₂ be the length of the latus rectum of y² = 8x, then:
(a) L₁ > L₂
(b) L₁ = L₂
(c) L₁ < L₂
(d) L₁/L₂ = √2
Answer : C

Question. If the tangent at the point P (x₁, y₁) to the parabola y² = 4ax meets the parabola y² = 4a (x + b) at Q and R, then the mid-point of QR is
(a) (x₁ + b, y₁ + b)
(b) (x₁ – b, y₁ – b)
(c) (x₁, y₁)
(d) (x₁ + b, y₁ – b)
Answer : C

Question. The normal at (2, 3/2) to the ellipse, x²/16 + y²/3 = 1 touches a parabola, whose equation is
(a) y² = – 104 x
(b) y² = 14 x
(c) y² = 26x
(d) y² = – 14x
Answer : A

Question. Tangents are drawn from O (origin) to touch the circle x² + y² + 2gx + 2fy + c = 0 at points P and Q. The equation of the circle circumscribing triangle OPQ is
(a) 2x² + 2 y² + gx + fy = 0
(b) x² + y² + gx + fy = 0
(c) x² + y² + 2gx + 2fy = 0
(d) None of these
Answer : B

Question. The radius of the circle passing through the foci of the ellipse x2/16 + y2/9 = 1, and having its centre at (0, 3) is
(a) 4
(b) 3
(c) √1/2
(d) 7/2
Answer : A

Question. Equation of the line passing through the points of intersection of the parabola x² = 8y and the ellipse x²/3 + y² = is :
(a) y – 3 = 0
(b) y + 3 = 0
(c) 3y + 1 = 0
(d) 3y – 1 = 0
Answer : D

Question. Equation of the largest circle with centre (1, 0) that can be inscribed in the ellipse x² + 4y² = 16, is
(a) 2x² + 2y² – 4x + 7 = 0
(b) x² + y² – 2x + 5 = 0
(c) 3 x² + 3 y² – 6x – 8 = 0
(d) None of these
Answer : C

Question. A circle bisects the circumference of the circle x² + y² – 2y – 3 = 0 and touches the line x = y and the point (1, 1). Its radius is :
(a) 3/√2
(b) 9/√2
(c) 4√2
(d) 3√2
Answer : B

Question. The angle subtended by the common tangent of the two ellipse (x - 4)²/25 + y²/4 = 1 and (x+1)²/1 + y²/4 = 1 at the origin is
(a) π/2
(b) π/4
(c) π/3
(d) π/6
Answer : A

Numeric Value Answer

Question. P(a,b) is a points in the first quadrant. Circles are drawn through P touching the coordinate axes, such that the length of common chord of these circle is maximum. If possible values of a/b is k₁ ± k₂ √2 then k₁ + k2 is equal to______.
Answer : 5

Question. Two tangents are drawn from a point (–2, –1) to the curve, y² = 4x. If a is the angle between them, then |tan a| is equal to:
Answer : 3

Question. S₁ and S₂ be the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, S₃ and S₄ be the foci of the conjugate hyperbola. If the area of the quadrilateral S₁ S₃ S₂ S₄ is A, then find A/13
.Answer : 2

Question. Two equal chords AB and AC of the circle x² + y² – 6x – 8y – 24 = 0 are drawn from the point A(√33 + 3,0) . Another chord PQ is drawn intersecting AB and AC at points R and S, respectively given that AR = SC = 7 and RB = AS = 3. The value of PR/QS is
Answer : 1

Question. If p and q be the longest and the shortest distance respectively of the point (–7, 2) from any point (a, b) on the curve whose equation is x² + y² -10x -14y - 51 = 0 and G.M. of p and q is 2√k , then value k is _______.
Answer : 11

Question. Tangents are drawn to the ellipse x²/9 + y₂/5 y = 1 at ends of latus rectum. The area of quadrilateral so formed is
Answer : 27

Question. A trapezium is inscribed in the parabola y² = 4x such that its diagonal pass through the point (1, 0) and each has length . 4/25 If the area of trapezium be P then [P/4] is equal to
Answer : 4

Question. The straight line y = mx + c (m > 0) touches the parabolas y² = 8 (x + 2) then the minimum value taken by c is
Answer : 4

Question. If the ratio of the area of equilateral triangles made of the common chord of the circles x² + y² = 4 and x² + y² – 8x + 4 = 0 and their respective pairs of tangents drawn from points on the positive x- axis is 57 + 24√3 : k then k is ________.
Answer : 9

Question. C is the centre of the hyperbola x²/4 - y²/1 = 1, and ' A' is any point on it. The tangent at A to the hyperbola meets the line x - 2y = 0 and x + 2y = 0 at Q and R respectively. The value of CQ.CR is equal to
Answer : 5